Opuscula Math. 41, no. 3 (2021), 335-379

Opuscula Mathematica

Multi-variable quaternionic spectral analysis

Ilwoo Cho
Palle E.T. Jorgensen

Abstract. In this paper, we consider finite dimensional vector spaces \(\mathbb{H}^n\) over the ring \(\mathbb{H}\) of all quaternions. In particular, we are interested in certain functions acting on \(\mathbb{H}^n\), and corresponding functional equations. Our main results show that (i) all quaternions of \(\mathbb{H}\) are classified by the spectra of their realizations under representation, (ii) all vectors of \(\mathbb{H}^n\) are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring \(M_n(\mathbb{C})\) of all \((n \times n)\)-matrices over the complex numbers \(\mathbb{C}\) has close connections with certain "non-linear" functional equations on \(\mathbb{H}^n\) up to the classification of (ii).

Keywords: the quaternions \(\mathbb{H}\), vector spaces \(\mathbb{H}^n\) over \(\mathbb{H}\), \(q\)-spectral forms, \(q\)-spectral functions.

Mathematics Subject Classification: 20G20, 46S10, 47S10.

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  • Ilwoo Cho (corresponding author)
  • St. Ambrose University, Department of Mathematics and Statistics, 518 W. Locust St., Davenport, Iowa, 52803, USA
  • Communicated by P.A. Cojuhari.
  • Received: 2020-10-07.
  • Revised: 2021-02-15.
  • Accepted: 2021-02-26.
  • Published online: 2021-04-19.
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Cite this article as:
Ilwoo Cho, Palle E.T. Jorgensen, Multi-variable quaternionic spectral analysis, Opuscula Math. 41, no. 3 (2021), 335-379, https://doi.org/10.7494/OpMath.2021.41.3.335

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